One thing you can say for epic heroes: they're not humble. (As if that's something to be proud of.) One of Gilgamesh's defining characteristics at the start of The Epic of Gilgamesh is his unwavering and excessive pride. There is nothing Gilgamesh doesn't think he can do; and, once he defeats Humbaba his pride only skyrockets—and we see that in IMAX 3D when he spurns the goddess Ishtar's proposition of love. Even when Enkidu dies and Gilgamesh goes in search of immortality, he finds it pretty hard to wrap his mind around the possibility that he is mortal like everybody else. So, again, we at Shmoop think "Gilgamesh" must be Sumerian for "cocky, swollen-headed king."
In The Epic of Gilgamesh, characters are able to swallow their pride when they know they have been beaten by the very best. Thus, in a paradoxical way, they are able to keep their pride intact.
Gilgamesh shifts from taking pride only in himself to taking pride in the city of which he is a member. This shows his shift away from complete self-centeredness, an important step in conquering the fear of death, and learning how to live.